Extends Frohman and Rannard's result to Seifert fiber spaces with singular surfaces.
problem Characterize essential surfaces in Seifert fiber spaces with singular surfaces.
method Extends Frohman and Rannard's approach to handle surfaces with singular fibers.
result Characterizes essential surfaces in Seifert fiber spaces with singular surfaces.
Study complex surfaces fibered over Teichmüller curves with Veech fibers.
problem Understanding the geometric and topological properties of complex surfaces.
method Compute invariants via monodromy action on fiber's mod-m homology. result Exact values of invariants for all known algebraically primitive Teichmüller curves.
Classifies fibering of state surfaces for various knot families.
problem Determining which state surfaces are fibered.
method Algebraic characterization of fibers from state graphs, decomposing graphs into planar components.
result Characterizes fibering for many families of state surfaces.
Study of Seifert fibered spaces using surface complexes.
problem Understanding the topology of Seifert fibered spaces.
method Defined surface complex for 3-manifolds, applied to Seifert fibered spaces.
result Surface complex of Seifert fibered spaces contains a subcomplex isomorphic to the curve complex of the base orbifold.
Constructs a family to handle unstable fibers on complex surfaces.
problem Handling unstable fibers on complex surfaces.
method Uses Teichmüller theory to construct a degenerating family over the moduli space.
result Any fibered complex surface with unstable fibers can be pulled back from the constructed family.
We show that a compact complex surface which fibers smoothly over a curve of genus >1 with fibers of genus >1 fibers holomorphically. We deduce an improvement of a result in [D Kotschick, Math. Research Letters, 5 (1998) 227-234], and a characterisation of fibered surfaces with zero signature.
This paper computes the number of fiberings for non-product manifolds.
problem Computing the number of fiberings for non-product manifolds.
method Analyzes specific manifolds and uses bundle properties.
result Fib(M)=2 for the Atiyah-Kodaira manifold and its finite covers, and Fib(M)=4 for an example. Describes 3-manifolds by families of singularly fibered surfaces.
problem Understanding the structure of 3-manifolds through fibered surfaces.
method Explicitly describes a fibration of the complement of a homogeneous braid.
result Each fiber intersects every cross-section of S3. Study shows quasipositive fiber surfaces can't be well-ordered.
problem Ordering quasipositive fiber surfaces is not well-ordered.
method Explicit construction of an infinite anti-chain.
result Quasipositive fiber surfaces are not well-quasi-ordered.
Unique incompressible surface in fibered 3-manifolds.
problem Finding unique incompressible surfaces in fibered 3-manifolds.
method Construction of a family of fibered hyperbolic 3-manifolds.
result The unique incompressible surface is the fibre itself.
Paper proves unbounded stabilization heights of fiber surfaces using Hopf invariant.
problem Unbounded stabilization heights of fiber surfaces.
method Alternative proof using Hopf invariant.
result Proves unbounded stabilization heights of fiber surfaces.
New research finds infinitely many hyperbolic knots not almost-fibered.
problem Understanding the properties of knots and their complements.
method Examined hyperbolic knots and their Seifert surfaces.
result Existence of infinitely many hyperbolic genus one knots that are not almost-fibered.
Study on surfaces in flag threefold with constraints on twistor fibers.
problem Understanding the arrangement and existence of twistor fibers in surfaces of specific bidegree.
method Analyzing surfaces of bidegree (1,d) in the flag threefold, proving existence and non-existence of twistor fibers.
result Existence and non-existence of surfaces containing specific numbers of twistor fibers, with improved results for d=2 and d=3.
New bounds found for minimal surfaces in hyperbolic 3-manifolds.
problem Bounding the maximum principal curvatures of minimal surfaces in hyperbolic 3-manifolds.
method Short argument using work of Uhlenbeck and families of fibered hyperbolic 3-manifolds.
result Uniform lower bound for maximum principal curvatures greater than one.
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each n≥3 we construct 4-manifolds E admitting at least n distinct fiberings pi:E→Σgi as a surface bundle over a surface with base and fiber both closed surfaces of negative Eule…
Study connects surface projections in fibered 3-manifolds.
problem Understanding surface projections in fibered 3-manifolds.
method Uniformly relates structure of surface projections across fibrations.
result Extends previous work to general cases.
We study principal curvatures of fibers and Heegaard surfaces smoothly embedded in hyperbolic 3-manifolds. It is well known that a fiber or a Heegaard surface in a hyperbolic 3-manifold cannot have principal curvatures everywhere less than one in absolute value. We show that given an upper bound on the genus of a minim…
Study introduces new indices for special fibered surfaces of genus 4.
problem Characterizing relatively minimal fibered surfaces of genus 4.
method Introduces Horikawa index and local signature for surfaces with unique trigonal structure.
result New indices provide insights into the structure of surfaces.
The study bounds and characterizes surfaces containing smooth conics and twistor fibers in a flag threefold.
problem Bounding and characterizing surfaces containing smooth conics and twistor fibers in a flag threefold.
method Analyzing the family of smooth conics and using algebraic properties to construct surfaces.
result The only smooth cases of surfaces containing infinitely many twistor fibers are of bidegree (1,1).
We characterize cutting arcs on fiber surfaces that produce new fiber surfaces, and the changes in monodromy resulting from such cuts. As a corollary, we characterize band surgeries between fibered links and introduce an operation called Generalized Hopf banding. We further characterize generalized crossing changes bet…
Classifies tight contact structures on specific Seifert fibered manifolds.
problem Classifying tight contact structures on Seifert fibered manifolds.
method Constructed contact structures using Legendrian surgery and used convex surface theory for the upper bound.
result Found the lower and upper bounds for tight contact structures.
Study ribbon concordance and minimal compressions, proving new results about fibered knots.
problem Understanding ribbon concordance and minimal compressions of surface homeomorphisms.
method Proving monotonicity of simplicial volume and dilatation under ribbon concordance, algorithmic enumeration of minimal compressions.
result Every fibered knot has only finitely many predecessors in the ribbon-concordance partial order.
Proves uniqueness of Lefschetz fibrations over discs based on singular fibers.
problem Classifying Lefschetz fibrations over the disc based on their singular fibers.
method Inspired by homological mirror symmetry and Karpov--Nogin's theorem, uses topological equivalence and fiber analysis.
result Lefschetz fibrations over discs are uniquely determined by their singular fibers.
Every Kauffman state σof a link diagram D(K) naturally defines a state surface S_σwhose boundary is K. For a homogeneous state σ, we show that K is a fibered link with fiber surface S_σif and only if an associated graph G'_σis a tree. As a corollary, it follows that for an adequate knot or link, the second and next-to-…
The paper studies mapping class groups of nontrivial S2 fiber bundles.
problem Analyzing the mapping class groups of nontrivial S2 fiber bundles. method Using generalizations of Dax invariants for embedded surfaces in 4-manifolds.
result Surjective homomorphisms from MCG(X) and MCG(X′) to Z∞ are shown. In "Tunnel one, fibered links", the second author showed that the tunnel of a tunnel number one, fibered link can be isotoped to lie as a properly embedded arc in the fiber surface of the link. In this paper, we analyze how the arc behaves under the monodromy action, and show that the tunnel arc is nearly clean, with t…
In this paper, we first classify singular fibers of proper C∞ stable maps of 3-dimensional manifolds with boundary into surfaces. Then, we compute the cohomology groups of the associated universal complex of singular fibers, and obtain certain cobordism invariants for Morse functions on compact surfaces with bo…
Minimal singular fibers found in nonorientable Lefschetz fibrations.
problem Finding the minimal number of singular fibers in nonorientable Lefschetz fibrations.
method Analyzing admissible nonorientable genus g Lefschetz fibrations over orientable surfaces.
result Existence of one singular fiber if and only if g ≥ 4 and h ≥ 1.
Translation distances in fibered 3-manifolds with boundary are bounded and grow with complexity.
problem Bounding translation distances in fibered 3-manifolds with boundary.
method Using essential surfaces with non-zero slope and analyzing their complexity.
result Translation distances are bounded and grow with complexity, supporting a conjecture.
Study bundles over surfaces with specific fibers, determining characteristic numbers and obstructions.
problem Characterizing bundles over surfaces with highly connected fibers.
method Analyzing smooth and topological bundles, providing necessary and sufficient conditions, and computing characteristic numbers.
result Determine characteristic numbers and divisibility constraints on signatures and genera for bundles of this type.
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided…
Study handles in sutured manifolds and knots, finding varied handle numbers and unique surfaces.
problem Understanding handle numbers and incompressible Seifert surfaces in sutured manifolds and nearly fibered knots.
method Extending Haken's Theorem, analyzing product annuli and disks, and examining specific knot types.
result Variety of handle numbers and unique incompressible Seifert surfaces in nearly fibered knots.
Study finds negatively curved spheres in elliptic surfaces and their modifications.
problem Finding negatively curved spheres in elliptic surfaces.
method Using elliptic fibrations with specific singular fibers.
result Identifies spheres with very negative self-intersections in elliptic surfaces and their modifications.
The study explores how surface diffeomorphisms of knots relate to their topological properties.
problem Understanding how properties of surface diffeomorphisms of knots relate to their topological properties.
method Examining both braid and fibered knot perspectives to explore the relationship between surface diffeomorphisms and knot properties.
result Properties of surface diffeomorphisms may relate to four-dimensional topological properties of knots, such as the slice genus.
It is known that there are surface bundles of arbitrarily high genus which have genus two Heegaard splittings. The simplest examples are Seifert fibered spaces with the sphere as a base space, three exceptional fibers and which allow horizontal surfaces. We characterize the monodromy maps of all surface bundles with ge…
Resolves singular fibers of a 5-manifold with S1 action.
problem Discrete singular fibers of a closed 5-manifold with S1 action. method Construction of resolution compatible with cyclic surface singularities.
result Compatibility proved between resolution and cyclic surface singularities.
Researchers create metrics collapsing to a line, detailing near-fiber behavior.
problem Constructing metrics on elliptic K3 surfaces that collapse to a line.
method Family of collapsing Ricci-flat Kähler metrics with bounded curvatures.
result Precise description of metric degeneration near singular fibers.
The paper studies elliptic surfaces and proves unique fibered structures.
problem Understanding the structure of elliptic fibrations and their mapping class groups.
method Analyzes smooth 4-manifolds and uses Néron-Lang Theorem. result Proves the uniqueness of fibered structures in elliptic fibrations.
Proof of existence for ideal triangulations that normalize fibers in certain 3-manifolds.
problem Existence of ideal triangulations that normalize fibers in specific 3-manifolds.
method Proof and algorithm construction for ideal triangulations.
result Existence of ideal triangulations that normalize fibers in certain 3-manifolds.
Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher. We conver…
Study of Einstein structures for surface group representations in specific Lie groups.
problem Understanding the geometric structures of representations in SO0(p,p+1). method Explicitly constructing fiber bundles and determining their diffeomorphism types.
result Many fiber bundles are trivial, revealing unique homotopy types.
Study Stein and Milnor fillings of links from surface singularities.
problem Comparing Stein and Milnor fillings of links from surface singularities.
method Analyzing the topology and obstructions of Stein fillings and Milnor fillings.
result Milnor fillings have bounded topology, while Stein fillings can be more varied.
New groups found that don't virtually algebraically fiber, related to mapping class group orbits.
problem Existence of finite orbits for higher Prym representations of the mapping class group.
method Study of surface-by-surface and surface-by-free groups.
result Existence of free-by-free and free-by-surface groups that do not algebraically fiber.
There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of tw…
The E_8-manifold has several natural framed link descriptions, and we give an efficient method (via `grapes') for showing that they are indeed the same 4-manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we sho…
The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
Dehn surgery on knots in 3-manifolds rarely results in surface fibered manifolds.
problem Understanding when Dehn surgery on knots in 3-manifolds results in surface fibered manifolds.
method Analyzing the fiberedness of Dehn surgeries on knots in 3-manifolds.
result Dehn surgery on knots in 3-manifolds is not surface fibered for all but a sparse set of slopes.
We will use immersed surfaces to study Seifert fibered surgery on Montesinos knots, and show that if q1−11+q2−11+q3−11≤1 then a Montesinos knot K(q1p1,q2p2,q3p3) admits no atoroidal Seifert fibered surgery.